Weighted Estimations from a Conjugate Operator* MARIUS MA - NTOIU** and RADU PURICE Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, Bucharest, RO-70700, Romania. e-mail: radu.purice@imar.ro (Received: 26 November 1999; revised version: 7 February 2000) Abstract. In this Letter we develop a general procedure leading from a Mourre-type estimation for a given self-adjoint operator H to a Hardy-type weighted inequality. We use this method in order to prove exponential decay for eigenvectors of a large class of perturbations of operators of convolution with bounded analytic functions. Mathematics Subject Classi¢cations (2000): 15A18, 26D15, 47A05. Key words: eigenfunction decay, Hardy inequality, conjugate operator, exponential decay. 1. Introduction In obtaining upper bounds on the decay of some given eigenfunctions of a linear operator H on L 2 R n , a very natural starting point seems to be a Hardy-type weighted estimation for H (see Theorem 2). Given a self-adjoint operator H acting in L 2 R n  and a real number E, one can consider the problem of obtaining esti- mations of the type w 1 f     W Cw 2 H  E f     1:1 for f in the domain of H and supported away from the origin, and with w 1 and w 2 some given weight functions. They allow one to deduce a given decay for f once H  Ef has a speci¢c decay. This kind of estimation has already appeared in such well-known papers as [1, 2, 7]. Some of the works on this subject [3, 4, 8, 9, 12, 13] may indicate that one can prove such estimations using a special type of positivity condition associated with the existence of a conjugate operator, an object which is very useful for the spectral analysis of the operator H. More precisely, we say that H satis¢es a Mourre estimation with respect to the conjugate operator A, at a real value E, when (denoting by E J H the spectral projection of H on an interval J * Research partially supported by the Swiss National Science Foundation and the grant CNCSU-13. ** Present address: DeÂpartement de Physique TheÂorique, Universite de GeneÁve; 32, bd. d'Yvoy; CH-1211 GeneÁve 4, Switzerland. email: marius.mantoiu@physics.unige.ch Letters in Mathematical Physics 51: 17^35, 2000. 17 # 2000 Kluwer Academic Publishers. Printed in the Netherlands.